By AK BOUSFIELD 1. Introduction. During the past two decades

34 Pages·1997·0.24 MB·English

HOMOTOPICAL LOCALIZATIONS OF SPACES
By A. K. BOUSFIELD
Abstract. For a map f of spaces, Dror Farjoun and the author have constructed an f-localization
functor, where a space Y is called f-local when map(f,Y) is an equivalence. This very general
construction gives all known idempotent homotopy functors of spaces. The main theorem of this
paper shows that f-localization functors always preserve fiber sequences of connected H-spaces
up to small error terms. For instance, the localization with respect to the nth Morava K-theory
preservessuchfibersequencesuptoerrortermswithatmostthreenontrivialhomotopygroups.This
implies,forexample,thataK(1)-homologyequivalenceofH-spacesmustinduceanisomorphism
of(cid:118)1-periodichomotopygroups.ResultsarealsoobtainedontheA-nullificationorA-periodization
functors, which are just the f-localization functors for the maps f from spaces A to points. Two
spaces are said to have the same nullity when they give the same nullification functors, and it is
shownthatarbitrarysetsofnullityclasseshavebothleastupperboundsandgreatestlowerbounds.
TheA-nullificationsofnilpotentPostnikovspacesarecompletelydetermined.
1. Introduction. During the past two decades, great progress has been
made toward a global understanding of stable homotopy theory, showing that
some major features arise “chromatically” from an interplay of periodic phe-
nomena arranged in a hierarchy (see [Ra]). These phenomena have been quite
effectivelyexposed using localizationsof spectra with respect to periodic homol-
ogy theories such as the Morava K-theories K(n) . We would like to similarly
expose periodic phenomena in unstable homotopy(cid:3)theory using localizations of
spaces.SomeencouragingprogressinthisdirectionhasbeenmadebyMahowald
and Thompson ([MT]),Dror Farjounand Smith([DFS]),the author([Bo 7]),and
others, and a powerful general theory of unstable homotopical localizations has
begun to emerge. In this paper, we investigate that theory and prove a general
fibration theorem showing, for instance, that K(n) -localizations preserve fiber
sequences of connected H-spaces up to error terms(cid:3)with at most three nontrivial
homotopy groups.
To describe our results, we first recall the very general notion of an f-
localization of spaces, and we initially work in the pointed homotopy category
Ho ofCW-complexes.Forafixedmapf: A Bofspaces,wesaythataspaceY
is f(cid:3)-local when f : map(B,Y) map(A,Y).!As shown by Dror Farjoun ([DF 1])
(cid:3)
and the author ([Bo 3, Corolla’ry 7.2]), each space X has a natural f-localization
X L X. The f-localization generalizes both the E -localization X X for a
f E
ho!mology theory E and the A-nullification X P (cid:3)X for a space A,!which may
A
(cid:3) !
ManuscriptreceivedMay31,1996.
ResearchsupportedinpartbytheNationalScienceFoundation.
AmericanJournalofMathematics119(1997),1321–1354.
1321
1322 A.K.BOUSFIELD
respectively be obtained using a suitable huge E -equivalence f and the trivial
map f: A . We give a brief general account (cid:3)of f-localizations in Section 2,
and refer t!he(cid:3)reader to [Bo 8], [Ca 1], and particularly to Dror Farjoun’s book
([DF 3]) for additional background information. We remark that Dror Farjoun’s
earlier paper ([DF 1]) helped to stimulate widespread interest in f-localizations.
For a map f and space A, we consider the localizationclass f consistingof
all maps giving the same local spaces asf, and the nullity classhAiconsisting of
all spaces giving the same null spaces as A. The collections Lochs iof localization
classes and Nuls of nullity classes have obvious partial orderings (see [Bo 7],
[DF 2], [DF 3]), and we prove that they are actuallysmall-completelargelattices
in the sense that their (small) subsets have greatest lower bounds and least upper
bounds (see 4.3 and 4.5). Moreover, we prove that each localization class f
has a best possible approximation by a nullity class A(f) such that P ahndi
A(f)
L have the same acyclic spaces (Theorem 4.4). ThushP iis related to L in the
f A(f) f
same way as Quillen’s plus-construction is related to the H ( ;Z)-localization
functor. For each map f, we also obtain a closed model cat(cid:3)eg(cid:0)ory structure for
spaces, where the “weak equivalences” are the L -equivalences (Theorem 4.6).
f
Thus each localization class f determines its own brand of homotopy theory.
Versions of this result have bheeni obtained by Dror Farjoun ([DF 3]), Hirschhorn
([Hi]), Smith, the author ([Bo 2, Appendix]), and others. Our present approach
actuallyshowstheexistenceof -localizationsandassociatedhomotopytheories
for many large classes of mapsF .
Our main result is a generaFl fibration theorem. For a map f of connected
spaces, we prove that the localization functors LS f and LfW preserve homotopy
fiber sequences up to error terms whose p-completions have at most three non-
trivial homotopy groups for each prime p (Main Theorem 9.7). This generalizes
a fibration theorem of Dror Farjoun and Smith ([DFS]) for the localization func-
tors L and L W 2, which in turn partially generalizes the fibration theorems of
S 2f f
[Bo 7] and [DFS] for the nullification functors PS A and PAW . These results all
depend on a key lemma (Lemma 5.3) which we originally proved in [Bo 7, 6.9].
The crux of our present proof is in Section 6, where we show that the homotopy
fiber of an LS f-equivalence of spaces is “almost” LS f-acyclic (Theorem 6.2), and
where we find a very convenientexpressionfor theLS f-errortermof a homotopy
fiber sequence (Theorem 6.4). Our main proof is completed in Section 9 after
we have determined the nullifications of nilpotent Postnikov spaces and of other
nilpotent “generalized polyGEMs” (see Theorems 7.5 and 8.8).
In Section 10, we develop general homological consequences of the preced-
ing work and show that K(n) -localizations, and other E -localizations, “almost”
preserve homotopy fiber sequ(cid:3)ences of H-spaces (Theore(cid:3)m 10.10). This also ap-
pliestovariouscohomologicallocalizationsincludingthosewithrespecttostable
cohomotopy theory (see 2.6 and 10.12). Finally, in Section 11, we introduce the
notion of a virtual E -equivalence, defined as a map of spaces : X Y such
that : (W X) (cid:3) (W Y) is an isomorphism for all sufficie(cid:30)ntly la!rge i. We
i E i E
(cid:30)(cid:3) (cid:25) ! (cid:25)
HOMOTOPICALLOCALIZATIONSOFSPACES 1323
find that the virtual E -equivalences are much more “durable” than ordinary
E -equivalences. For in(cid:3)stance, in a map of fiber sequences, if any two of the
co(cid:3)mponent maps is a virtual E -equivalence, then so is the third (Theorem 11.4).
We show that each E -equivale(cid:3)nce of H-spaces is a virtual E -equivalence (The-
orem 11.3), and each(cid:3)virtualE -equivalenceof spaces induce(cid:3)s anE -equivalence
of sufficiently highly connect(cid:3)ed covers (Theorem 11.7). We dedu(cid:3)ce that if an
H-space is E -acyclic, then so are all of its connected covers and all of its Post-
nikov section(cid:3)s (Theorem 11.9). Turning to K-theory, we show that the virtual
K p -equivalences (or virtual K(1) -equivalences) of spaces are the same as the
(cid:118)=1(cid:3) ( ;Z p)-equivalences (Theo(cid:3)rem 11.11), and conclude, for instance, that
1(cid:0)
each(cid:25)(cid:3)K(cid:0)p -=equivalence of H-spaces is a (cid:118) 1 ( ;Z p)-equivalence (Corollary
1(cid:0)
11.12).=Th(cid:3)is generalizes a result of Thompson(cid:25)([(cid:3)T(cid:0)h], [=Bo 7, 11.9]), and is needed
fortheauthor’ssubsequentworkonK p -localizationsand(cid:118) -periodizations.We
1
also obtain some results on virtual K(=n)(cid:3)-equivalences of spaces for n 1. We
showthatifamapofspacesisa(cid:118)-period(cid:3)ichomotopyequivalencefor1> j n,
j
then it is a virtual K(n) -equivalence (Theorem 11.13). This implies, fo(cid:20)r ex(cid:20)am-
ple, that if X is a space(cid:3)with trivial (cid:118)-periodic homotopy groups for 1 j n,
j
then the Postnikov map X Pn+1X is a K(n) -equivalence (Corollary(cid:20)11(cid:20).14).
This should help to make K!(n) X more accessi(cid:3)ble, and extends a similar result
of Hopkins, Ravenel, and Wilso(cid:3)n ([HRW]) for infinite loop spaces.
This paper generalizes the fundamental results of Dror Farjoun and Smith
([DFS]), and we have benefited from their ideas.
We work simplicially so that “space” means “simplicial set.” However, to
make the presentation more accessible, we frequently work in the pointed homo-
topy category Ho .
(cid:3)
2. The basic homotopical localization theory. In this section, we recall
the basic theory of f-localizations of spaces and discuss some general examples.
We refer the reader to [Bo 8], [Ca 1], and [DF 3] for additional background
informationandresults.Athoroughaccountofthebasictheoryisbeingdeveloped
by Hirschhorn ([Hi]) in a general model category setting.
For pointed spaces X, Y Ho , let map (X,Y) Ho and map(X,Y) Ho
respectivelydenotethepointe2dand(cid:3)unpointed(cid:3) mappi2ngsp(cid:3)acesfromX toafi2bran(cid:3)t
representative for Y, and recall that map (X,Y) = [X,Y]. For a map f: A B
0
(cid:24)
and space Y in Ho , we consider th(cid:25)e ortho(cid:3)gonality conditions: !
(cid:3)
(H1) f : [B,Y] = [A,Y];
(cid:3) (cid:24)
(H2) f : map (B,Y) map (A,Y);
(cid:3)
(cid:3) ’ (cid:3)
(H3) f : map(B,Y) map(A,Y).
(cid:3)
’
It is easy to show
LEMMA 2.1. (H3) (H2) (H1)and,whenY isconnected,(H2) (H3).
) ) ,
1324 A.K.BOUSFIELD
We adopt (H3) as our main orthogonality condition in Ho . For a fixed map
f: A B in Ho , a space Y Ho is called f-local when(cid:3)f : map(B,Y) =
(cid:3) (cid:24)
map(A!,Y); a map(cid:3) u: X X 2in H(cid:3)o is called an f-local equivalence when
0
u : map(X ,Y) = map(X,!Y) for each f(cid:3)-local space Y; and a map u: X X is
(cid:3) 0 (cid:24) 0
called an f-localization when it is an f-local equivalence to an f-local sp!ace X .
0
By Lemma 2.1, an f-localization u: X X is an initial example of a map from
0
X to an f-local space in Ho , and is a te!rminal example of an f-local equivalence
out of X in Ho . Thus the(cid:3)f-localizations are unique up to equivalence in Ho ,
and by [Bo 3, C(cid:3)or. 7.2] or [DF 1], we have (cid:3)
THEOREM 2.2. For each map f: A B and space X in Ho , there exists an
f-localizationofX. ! (cid:3)
Hence, there is an idempotent functor L : Ho Ho giving a natural f-
f
localization u: X L X for X Ho . (cid:3) ! (cid:3)
f
! 2 (cid:3)
2.3. The functor L on spaces. The idempotent functor L : Ho Ho
f f
is actually induced from a coaugmented functor L : on the(cid:3)c!ategory(cid:3)
f
of spaces (i.e. simplicial sets). Roughly speaking, fSor!X S , L X may be
f
cSonstructed from X by expressing f as an inclusion of spaces2AS B and taking
all possible pushouts from the pairs (D n,D˙n) (B,A) with n (cid:26)0 and the pairs
(D n,D n) with 0 k n 0, where D n den(cid:2)otes the standar(cid:21)d n-simplex with
k
boundary D˙n and(cid:20)kth h(cid:20)orn D>n. This construction is continued over an appropriate
k
section of ordinals to achieve the extension property with respect to the above
pairs and to create L X as a colimit. More elaborate versions of this construction
f
in [Bo 5], [Bo 7], and [DF 3] produce a functor L which is simplicial in the
f
sense of Quillen ([Qu, II.1]) with L ( ) = for a point .
f
We refer the reader to Casacubert(cid:3)a and(cid:3)Peschke ([C(cid:3)P]) for an analysis of the
f-localization in the illuminating basic case where f is a self-map of S1; we now
turn to some other important general examples.
2.4.Nullifications. ForaspaceA Ho ,thelocalizationwithrespecttothe
trivial map f: A is called the A-nu2llifica(cid:3)tion or A-periodization; the functor
L is denoted by!P (cid:3); the f-local spaces are called A-null or A-periodic; and the
f A
f-local equivalences are called A-periodic equivalences or P -equivalences. For
A
connectedspacesA,Y Ho ,notethatY isA-nullifandonlyifmap (A,Y) = .
(cid:24)
Thus the Sn+1-nullifica2tion (cid:3)functor on Ho is equivalent to the nth(cid:3) Postniko(cid:3)v
functor. Many other interestingnullification(cid:3)sare discussed in [Bo 7], [Bo 8], [Ca
2], [Ch], [DF 3], and [Ne].
2.5.Homologicallocalizations. Fora spectrumE,theE -localizationfunc-
tor ( ) : Ho Ho of [Bo 2] may be viewed as an f-loca(cid:3)lization for a huge
E
E -equivalenc(cid:3)e!f. For(cid:3)instance, we may use the map f: A B obtained
by(cid:3)wedging representatives A B of all isomorph_is(cid:11)m(cid:11)cla!sse_s(cid:11)of(cid:11)inclusions
f (cid:11) (cid:26) (cid:11)g(cid:11)
HOMOTOPICALLOCALIZATIONSOFSPACES 1325
of pointedspaces withE (B ,A ) = 0 and with cardinality#B #E (pt) where
#B denotes the number(cid:3)of(cid:11)non(cid:11)degenerate simplices of B . (cid:11)Th(cid:20)is fo(cid:3)llows since
the(cid:11)f-localization map u: X L X is an E -equivalence by(cid:11) its construction and
f
since L X is E -local by [Bo!2, Lemma 11.(cid:3)3].
f
(cid:3)
2.6. Cohomological localizations. Let E be a spectrum whose modp ho-
motopy groups (E p) are all finite for each primep. Then, following [Bo 4] or
[Ho]asexplaine(cid:25)d(cid:3)be=low,thereexistsa spectrumG suchthattheG -equivalences
are the same as the E -equivalences for spectra and hence for sp(cid:3)aces. Thus by
(cid:3)
2.5, there is an E -localization functor of the form L for spaces. To construct
(cid:3) f
G, recall that the (E p) -equivalences are the same as the c(E p) -equivalences
(cid:3)
where c(E p) is the B=rown-Comenetz ([BC]) dual of E p. Thus=wh(cid:3)en the groups
E are al=l Ext-complete (i.e. when Hom(Q, E) = 0== Ext(Q, E)), we may
u(cid:25)s(cid:3)e G = c(E p); and when the groups E(cid:25)a(cid:3)re not all Ext-com(cid:25)p(cid:3)lete, we may
p
use G = H_Q = c(E p) where HQ is the(cid:25)ra(cid:3)tional Eilenberg-MacLane spectrum.
p
We do not k_no_w wh=ether localizations of spaces exist with respect to arbitrary
cohomology theories, although they do for all ordinary cohomology theories by
[Bo 1].
3. Localizations with respect to classes of maps. The notion of an f-
localization of spaces for a single map f can obviously be extended to that of an
-localization for a class = f : A B of maps in Ho . In particular,
aFspace Y Ho is calleFd f-lo(cid:11)cal (cid:11)wh!en f(cid:11)g:(cid:11)map(B ,Y) = (cid:3)map(A ,Y) for
(cid:3) (cid:24)
each f 2; a m(cid:3)ap u: X FX in Ho is ca(cid:11)lled an (cid:11)-local equivalen(cid:11)ce when
0
u : m(cid:11)ap2(XF,Y) = map(X,Y!) for each (cid:3)-local space YF; and a map u: X X is
(cid:3) 0 (cid:24) 0
called an -localization when it is anF -local equivalence to an -loca!l space
X . Let (F ) denote the class of -locFal spaces in Ho . When F= f is a
0
(small)sLetFofmapsinHo ,thereisFasinglemapf = f(cid:3) suchthaFt (ff)(cid:11)=g(cid:11)( ),
and the f-localizations in(cid:3)Ho immediately give -l_o(cid:11)ca(cid:11)lizations. InLthis seLctiFon,
we develop machinery showi(cid:3)ng that many large Fclasses of maps = f can
similarly be replaced by single maps f with (f) = ( ). This Fwill gfe(cid:11)nge(cid:11)ralize
the prototypical example of E -localizationsL(2.5) wLheFre the class of all E -
equivalences is replaced by a si(cid:3)ngle huge E -equivalence. Our main application(cid:3)s
of this machinery will be given in Section 4(cid:3). We remark that in every case where
we are able to show the existence of -localizations in Ho , we are also able to
replace by a single map f. We shalFl need (cid:3)
F
3.1. Coherent functors. Let Sets be the category of pointed sets; let
(cid:3) S(cid:3)
be the category of pointed spaces (i.e. pointed simplicial sets); and let != 2
be the usual category of maps in . For such a map f: A B, writeS#(cid:3)f fSo(cid:3)r
the number of nondegenerate simplSic(cid:3)es of A B, and call f : !A B a submap
0 0 0
of f (denoted by f f) when A A, B _B, and f = f A.!Call a functor
0 0 0 0 0
(cid:26) (cid:26) (cid:26) j
1326 A.K.BOUSFIELD
T: ! Sets b-coherent for an infinite cardinal number b when each f !
hasST(cid:3)(!f) = co(cid:3)lim T( ) for ranging over the submaps of f with # b. T2hSi(cid:3)s
(cid:30) (cid:30) (cid:30) (cid:30) (cid:20)
is equivalent to saying that T preserves colimits of diagrams in ! indexed by
directedsetshavingupperboundsfortheirsubsetsofcardinality Sb.Notethatif
(cid:20)
T is b-coherent, then it is b-coherent for all b b. Call a functor T: ! Sets
0 0
coherent when it is b-coherent for sufficiently(cid:21)large b. For such a T,Sca(cid:3)l!l a map(cid:3)
f ! T-acyclic when T(f) = . For instance, for a spectrum E, the relative
2 S(cid:3) (cid:3)
homology functor E : ! Sets is coherent and the E -acyclic maps are the
E -equivalences. (cid:3) S(cid:3)! (cid:3) (cid:3)
(cid:3)
LEMMA 3.2. IfT: ! Sets isacoherentfunctor,thenthereexistsaninfinite
S(cid:3)! (cid:3)
cardinal number d such that for each T-acyclic g ! and each g with
2 S(cid:3) (cid:18) (cid:26)
# 2d,thereexistsaT-acyclic ! with gand# 2d.
(cid:18) (cid:20) (cid:18) 2 S(cid:3) (cid:18) (cid:26) (cid:18) (cid:26) (cid:18) (cid:20)
Proof. Assume that T is b-coherent and let d be a cardinal such that b d
(cid:20)
and #T( ) d for all ! with # b. Then for each f ! with
#f 2d,(cid:30)the(cid:20)re are at mo(cid:30)st2(2d)Sb(cid:3)= 2d sub(cid:30)m(cid:20)aps f with # b2, anSd(cid:3)hence
(cid:20) (cid:30) (cid:26) (cid:30) (cid:20)
#T(f) d 2d = 2d. Given a T-acyclic map g ! and g with # 2d,
each el(cid:20)emen(cid:1)t x T( ) maps trivially to T( ) fo2r sSom(cid:3) e (cid:18) (cid:26)g with (cid:18) (cid:20)and
0 0 0
# 2d. Hence2, the(cid:18)re is a transfinite increa(cid:18)sing sequenc(cid:18)e (cid:26) (cid:18) (cid:26) (cid:18)
0
(cid:18) (cid:20)
= =
0 1 +1
(cid:18) (cid:18) (cid:26) (cid:18) (cid:26) (cid:1)(cid:1)(cid:1)(cid:26) (cid:18)(cid:21) (cid:26) (cid:18)(cid:21) (cid:26) (cid:1)(cid:1)(cid:1)(cid:26)(cid:18)(cid:13) (cid:18)
of submaps of g indexed through the first ordinal of cardinality greater than
b, where each T( ) T( ) is trivial, where # (cid:13) 2d for each , and
+1
where = (cid:18)(cid:21) !for e(cid:18)a(cid:21)ch limit ordinal (cid:18)(cid:21). S(cid:20)ince T is b-co(cid:21)he(cid:20)re(cid:13)nt, we
deduce(cid:18)t(cid:12)hat T[((cid:11)<)(cid:12)=(cid:18)(cid:11)colim T( ) = and t(cid:12)ak(cid:20)e (cid:13)= .
(cid:18)(cid:13) (cid:21)<(cid:13) (cid:18)(cid:21) (cid:3) (cid:18) (cid:18)(cid:13)
For a coherent functor T: S! Sets , let (T) denote the class of all maps
in Ho represented by T-acyclic(cid:3)m!aps in(cid:3) . A
(cid:3) S(cid:3)
THEOREM 3.3. If T: ! Sets is a coherent functor, then (T) = (f)
for some wedge f of T-aScy(cid:3)c!lic map(cid:3)s. Hence, (T)-localizatonsLexAist in HoLand
(cid:0) (cid:1)
aregivenbyf-localizations. A (cid:3)
Proof.Letd beaninfinitecardinalgivenbyLemma3.2.TheneachT-acyclic
map in is the colimit of the directed system of allT-acyclic submaps
with(cid:30)# S(cid:3)2d, and thus is weakly equivalent to a homotopy colimit of(cid:18)th(cid:26)es(cid:30)e
submap(cid:18)s(cid:20)by[BK,p.332](cid:30).Hence, ( (T))= ( )where isasetcontaininga
representativeofeachisomorphismLcAlassofTL-acWyclicmapsW in with# 2d.
Thus we may let f be the wedge of all maps in . (cid:30) S(cid:3) (cid:30) (cid:20)
W
HOMOTOPICALLOCALIZATIONSOFSPACES 1327
NotethatthistheoremgivesanotherproofoftheexistenceofE -localizations
(cid:3)
in Ho (see 2.5) using the relative homology functor E : ! Sets . Before
turning(cid:3) to our main applications, we must formulate a no(cid:3)ncoSn(cid:3)n!ected W(cid:3)hitehead
theorem (Lemma 3.4) and derive a partial converse (Theorem 3.5) to the above
theorem.LetHo! be thehomotopycategoryobtainedby invertingthetermwise
S(cid:3)
weak equivalence in ! (see e.g. [BF, A.3]).
S(cid:3)
LEMMA 3.4. Amapofpointedspaces : X Y isaweakequivalenceifand
onlyifthenaturalfunctionh : X [i , (cid:30)]ison!toforn 0where[i , ]consists
n 0 n n
ofthemorphismsfromi : D˙n (cid:25) !D n (cid:30)to inHo!.(cid:21) (cid:30)
n
[(cid:3) (cid:26) [(cid:3) (cid:30) S(cid:3)
Proof. We can assume that is a fibration of the fibrant spaces. Then the
surjectivityof the functionsh is(cid:30)equivalentto the rightliftingpropertyof with
n
respect to the map i , and this is equivalent to the weak equivalence prop(cid:30)erty.
n
THEOREM 3.5. For each map of pointed spaces f: A B, there exists a co-
!
herent functor T : ! Sets whose acyclic maps are the f-local equivalences
f
in . S(cid:3)! (cid:3)
S(cid:3)
Proof. By Lemma 3.4, a map of pointed spaces : X Y is an f-local
equivalence if and only if h : L X [i ,L ] is o(cid:30)nto fo!r each n 0. For
n 0 f n f
an infinite cardinal number b (cid:25)#f, eac!h space X(cid:30) has L X = colim L X(cid:21) where
f f
X are the subspaces of X(cid:21)of cardinality b. Thus a suitable fu(cid:11)ncto(cid:11)r T is
f
f (cid:11)g(cid:11) (cid:20)
T ( ) = 1 [i ,L ] im h .
f n f n
(cid:30) (cid:30)
n_=0 (cid:14)
Note. The definitions and results of this section have obvious versions for
unpointed spaces with in place of .
S S(cid:3)
4. The lattice of localization functors and closed model category struc-
tures. Using the preceding machinery, we now prove several fundamental re-
sults on homotopical localizations: that the possible localization functors form a
small-completelarge lattice;that each localizationfunctor has a best possible ap-
proximationby a nullification;that the nullityclassesalso form a small-complete
large lattice; and that each localization functor determines a closed model cate-
gory structurefor spaces, and thus determinesits own brand of homotopy theory.
For a class of maps in Ho , we let ( ) denote the class of all -local
equivalences inFHo . In general, (cid:3) ( )EanFd F
(cid:3) F (cid:26) E F
1328 A.K.BOUSFIELD
LEMMA 4.1. Ifthefollowingconditionsaresatisfied,then = ( ):
F E F
(i) foreachspaceX Ho ,thereexistsamapX X in suchthatX is
0 0
-local; 2 (cid:3) ! F
F
(ii) eachequivalenceinHo belongsto ;
(cid:3) F
(iii) if a composition gf is defined in Ho and if any two of f,g, gf are in ,
thensoisthethird. (cid:3) F
Proof. A map f: X Y in Ho induces a map f : X Y , where X X
0 0 0 0
andY Y are -locali!zationsgive(cid:3)nby(i).Iff ( ),the!nf isanequiva!lence
0 0
in Ho!and hencFe f by (ii) and (iii). 2E F
(cid:3) 2F
THEOREM 4.2. For a set f of maps in Ho , there exists a map f such that
(f)= (f ). f (cid:11)g(cid:11) (cid:3)
E (cid:11)E (cid:11)
T
Proof.ByTheorem3.5,foreach ,thereisacoherentfunctorT : ! Sets
f
(cid:11) (cid:11) S(cid:3)! (cid:3)
with (T ) = (f ). These combine to give a coherent functor T: ! Sets
f
with TA( )(cid:11)= ET ((cid:11) ), where (T) = (f ). By Theorem 3.3, therSe(cid:3)i!s a map(cid:3)
f
f (cid:30)(f )_w(cid:11)ith(cid:11) (cid:30)(f)= A (f ) .(cid:11)SEinc(cid:11)e f is an f -equivalence for each ,
T
so2are(cid:11)thEe f(cid:11)-localizaEtion maEps,(cid:11)anEd (cid:11) (f ) = (cid:11)(f ) by Lemma 4.1. (cid:11)
T (cid:0)T (cid:1)
E (cid:11)E (cid:11) (cid:11)E (cid:11)
(cid:0)T (cid:1) T
4.3. The lattice of localization functors. Two maps f and g in Ho give
equivalentfunctorsL L ifandonlyif (f) = (g).Theresultingequiva(cid:3)lence
f g
classes f form a pa’rtially ordered collLection LLocs, where f g means
(f) h i(g) or equivalently (f) (g). Each (small) seth if(cid:20) h iin Locs
Lhas a(cid:27)leaLst upper bound fE and(cid:26)haEs a greatest lower bounfdh (cid:11)ifg(cid:11)given by
Theorem 4.2. Hence, Lochs_i(cid:11)s(cid:11)aismall-complete large lattice. For hfi g in
Locs, the idempotent localization functors L and L on Ho arehreila(cid:20)tedh biy a
f g
canonical transformation L L giving L L L . (cid:3)
f g g f g
A space X is called L -a!cyclic or f-acyclic’when L X , and X is called
f f
P -acyclic or killedby A when P X . ’ (cid:3)
A A
’ (cid:3)
THEOREM 4.4. For each map f in Ho , there exists a space A(f) Ho such
thatP andL havethesameacyclicsp(cid:3)aces. 2 (cid:3)
A(f) f
Proof. Let T : ! Sets be a coherent functor whose acyclic maps are the
f
f-equivalences.ThenS(cid:3)b!y Lemm(cid:3) a 3.2, there exists an infinite cardinal 2d such that
eachf-acyclicspaceX isthecolimitofadirectedsystemoff-acyclicsubspacesof
cardinality 2d. Thus A(f) exists as a wedge of representatives of isomorphism
classes of p(cid:20)ointed f-acyclic spaces of cardinality 2d.
(cid:20)
For example, if L is the H ( ;Z)-localization functor, then P is the
f A(f)
Quillen plus-construction functor(cid:3)by(cid:0)[Ca 1] or [DF 3].
HOMOTOPICALLOCALIZATIONSOFSPACES 1329
4.5. The lattice of nullity classes. Two pointed spaces X, Y Ho give
equivalent functors P P if and only if the X-null spaces are t2he sa(cid:3)me as
X Y
the Y-null spaces. The re’sulting equivalence classes X are called nullityclasses
(see [Bo 7, 9], [Bo 8], [Ch], [DF 2], or [DF 3]), ahndiform a partially ordered
collection Nxuls, where X Y means that the Y-null spaces are X-null or
equivalently that X is khillied(cid:20)byh Yi. There is an inclusion Nuls Locs where
X is identified with X . For each f Locs, Theore(cid:26)m 4.4 gives a
hgreiatest member A(f)h !Nul(cid:3)siwith A(f) h i 2f . Thus each set X in
Nuls has a least hupperib2ound X h andih(cid:20)ashaigreatest lower bofuhnd(cid:11)igg(cid:11)iven
by 4.3 and 4.4. Hence, Nuls is ha_s(cid:11)ma(cid:11)lil-complete large lattice. In addition, Nuls
has the obvious finite smash products. For V W in Nuls, the idempotent
nullificationfunctorsP andP onHo arerhelaite(cid:20)dbhyaicanonicaltransformation
V W
P P giving P P P . (cid:3)
V W W V W
!Finally,weshowthat’eachmapf inHo determinesaclosedsimplicialmodel
categorystructureon ,andthusdetermin(cid:3)esahomotopytheory.Versionsofthis
result have been obtaSin(cid:3)ed by Dror Farjoun ([DF 3]), Hirschhorn ([Hi]), Smith,
the author ([Bo 2, Appendix]), and others. We call a map in S an f-trivial
cofibration when it is both an f-local equivalence and a cofib(cid:30)ration(cid:3), and we call
an f-fibration when ithas the rightliftingpropertyforthef-trivialcofibrations.
(cid:30)
THEOREM 4.6. For each map f in Ho , the simplicial category of pointed
spaceshasaclosedsimplicialmodelcateg(cid:3)orystructurewith“weakeSqu(cid:3)ivalences,”
“fibrations,” and “cofibrations” respectively defined as f-local equivalences, f-
fibrations,andordinarycofibrations.
Proof. First note that a map is an ordinary trivial fibration if and only if
it is both an f-local equivalence (cid:30)and f-fibration, where the “if” part follows by
factoring as ji for a cofibration i and trivial fibration j, then deducing that i is
an f-trivia(cid:30)l cofibration, and concluding that is a retract of j. The theorem now
followsfromLemma 4.7 belowand adirect(cid:30)checkofQuillen’sconditionSM7(b)
([Qu]).
LEMMA 4.7. Eachmap : X Y in canbefactoredas = jiforanf-local
equivalenceiandf-fibration(cid:30)j. ! S(cid:3) (cid:30)
Proof. Let T : ! Sets be a coherent functor whose acyclic maps are
f
the f-local equivalenSce(cid:3)!s. Then(cid:3)by Lemma 3.2, there exists an infinite cardinal
2d such that each f-trivial cofibration is the colimit of a directed system of
f-trivial subcofibrations of cardinality (cid:11) 2d, and hence is equivalent to the
homotopy colimit of these subcofibratio(cid:20)ns. Consequently,(cid:11)an ordinary fibration
of fibrant spaces is an f-fibration if and only if has the right lifting propert(cid:18)y
for the f-trivial cofibrations of cordinality 2d. T(cid:18)hus, by a transfinite inductive
construction, we may factor the composite(cid:20)of : X Y with e: Y Ex Y to
1
give e = ji for an f-local equivalence i an(cid:30)d f-fib!ration j, where(cid:26)e is Kan’s
0 0 0 0
(cid:30)
1330 A.K.BOUSFIELD
weak equivalence to a fibrant space Ex Y. A pullback now gives the required
1
factorization of .
(cid:30)
Note. Theorem 4.6 and its proof can immediately be modified to show that
a map f actually determines a closed simplicial model category structure on the
category of unpointed spaces.
S
5. Acyclic spaces and their loop spaces. We now let f: A B be a fixed
map in Ho and consider thef-localizationu: X L X of a space!X. The reader
f
should kee(cid:3)p in mind the case of a (co)homologic!al localization (see 2.5 and 2.6)
which will be studied more fully in Sections 10 and 11. We say that a space X
is f-acyclic when L X , or equivalently by Theorem 4.4 when P X .
f A(f)
The f-acyclic point’ed(cid:3)spaces are closed under homotopy colimits and’un(cid:3)der
fiberextensions,butarenotclosedundermosthomotopyinverselimits.However,
the following key theorem will show that the loopspace of an f-acyclic H-space
is “almost” f-acyclic. A space M Ho is called a GEM when M is connected
2 (cid:3)
and M 1 K( M,n) with M abelian.
n 1
’ n=1 (cid:25) (cid:25)
Q
THEOREM 5.1. If Y Ho is a connected f-acyclic H-space, then LS fY is a
GEM,asarethecompon2entso(cid:3)fL W Y.
f
Thiswillbeprovedin5.6aftersomepreliminaries,andwillimplytherelated
results of [Bo 7], [Bo 8], and [DF 5]. For each space Y Ho , the loop space
W LS fY isf-local,andthusW u: W Y W LS fY inducesama2p : (cid:3)LfW Y W LS fY.
A fundamental result of Dror Farjo!un ([DF 3]) and the autho(cid:21)r ([Bo 7]!) is
THEOREM 5.2. IfY Ho isaconnectedspace,then : LfW Y W LS fY.
2 (cid:3) (cid:21) ’
Thus, in Theorem 5.1, it suffices to show thatLS fY is a GEM. Our main tool
from [Bo 7, Cor. 6.9] will be
KEY LEMMA 5.3. For connected spaces X, Y Ho , if map (X,Y) is homo-
topicallydiscreteandif Y actstriviallyon [X,Y2], then(cid:3) theinclu(cid:3)sionX SP X
1 1
inducesanequivalence m(cid:25)ap (SP X,Y) map (X,Y). (cid:26)
1
(cid:3) ’ (cid:3)
The infinite symmetric product SP X is a GEM with SP X = H (X;Z)
1 1 (cid:24)
by Dold-Thom, and we may obtain other GEMs by (cid:25)(cid:3) (cid:3)
e
LEMMA 5.4. IfM Ho isaGEM,thensoareitshomotopyretracts.
2 (cid:3)
Proof. For a homotopy retractionr: M N with homotopy fiber i: F M,
choosea map h: M K( F,n) such th!ath i : F = F, and deduc!e that
n n
(cid:24)
F is a GEM with M! F (cid:25)N. Then reverse th(cid:3)e(cid:3)ro(cid:25)le(cid:3)s of F(cid:25)(cid:3)and N to conclude
Q
that N is a GEM. ’ (cid:2)